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SUMMARY:Takumi Murayama (Princeton University)
DTSTART:20201008T190000Z
DTEND:20201008T203000Z
DTSTAMP:20260423T021430Z
UID:FOTR/25
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/FOTR/25/">Gr
 othendieck's localization problem</a>\nby Takumi Murayama (Princeton Unive
 rsity) as part of Fellowship of the Ring\n\n\nAbstract\nLet $A\\to B$ be a
  flat local map of noetherian complete local rings. Using Hironaka's resol
 ution of singularities Grothendieck and Dieudonné showed that if the clos
 ed fiber of the map $A\\to B$ is Cohen-Macaulay and if $A$ is of equal cha
 racteristic zero\, then all the fibers of the map are Cohen-Macaulay. Thre
 e decades later\, Avramov and Foxby showed that the same statement holds w
 ithout the characteristic assumption on A. Grothendieck's localization pro
 blem asks whether a similar statement holds with Cohen-Macaulayness replac
 ed by other local properties of noetherian local rings. We solve Grothendi
 eck's localization problem for all sufficiently well-behaved properties of
  noetherian local rings. Our proof provides a uniform treatment of previou
 sly known special cases of Grothendieck's problem\, in particular giving a
  new proof of Avramov and Foxby's result. As an application\, we show that
  if the closed fibers of a flat morphism of algebraic varieties are smooth
 \, then all fibers are smooth.\n
LOCATION:https://researchseminars.org/talk/FOTR/25/
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